Preliminaries

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library(MetricGraph)
library(ggplot2)
library(reshape2)
library(plotly)

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source(here::here("functionality.R"))

1 Finite element basis functions on a metric graph

Let each edge \(e\in\mathcal{E}\) be subdivided into \(n_{e}\geq 2\) regular segments of length \(h_{e}\), and be delimited by the nodes \(0 = x_0^{e},x_1^{e},\dots,x_{n_{e}-1}^{e}, x_{n_{e}}^{e} = \ell_{e}\). For each \(j = 1,\dots,n_{e}-1\), we consider the following standard hat basis functions \[\begin{equation*} \varphi_j^{e}(x)=\begin{cases} 1-\dfrac{|x_j^{e}-x|}{h_{e}},&\text{ if }x_{j-1}^{e}\leq x\leq x_{j+1}^{e},\\ 0,&\text{ otherwise}. \end{cases} \end{equation*}\] For each \(e\in\mathcal{E}\), the set of hat functions \(\left\{\varphi_1^{e},\dots,\varphi_{n_{e}-1}^{e}\right\}\) is a basis for the space \[\begin{equation*} V_{h_{e}} = \left\{w\in H_0^1(e)\;\Big|\;\forall j = 0,1,\dots,n_{e}-1:w|_{[x_j^{e}, x_{j+1}^{e}]}\in\mathbb{P}^1\right\}, \end{equation*}\] where \(\mathbb{P}^1\) is the space of linear functions on \([0,\ell_{e}]\). For each vertex \(v\in\mathcal{V}\), we define \[\begin{equation*} \mathcal{N}_v = \left\{\bigcup_{e\in\left\{e\in\mathcal{E}_v: v = x_0^e\right\}}[v,x_1^e]\right\}\bigcup\left\{\bigcup_{e\in\left\{e\in\mathcal{E}_v: v = x^e_{n_e}\right\}}[x^e_{n_e-1},v]\right\}, \end{equation*}\] which is a star-shaped set with center at \(v\) and rays made of the segments contiguous to \(v\). On \(\mathcal{N}_v\), we define the hat functions as \[\begin{equation*} \phi_v(x)=\begin{cases} 1-\dfrac{|x_v^{e}-x|}{h_{e}},&\text{ if }x\in\mathcal{N}_v\cap e \text{ and }e\in\mathcal{E}_v,\\ 0,&\text{ otherwise}, \end{cases} \end{equation*}\] where \(x_v^e\) is either \(x_0^e\) or \(x_{n_e}^e\) depending on the edge direction and its parameterization. See Arioli and Benzi (2018) for more details. Figure 1 shows an illustration of the basis function system \(\{\varphi_j^e, \phi_v\}\) on the tadpole graph (in black). Standard hat functions associated with internal edge nodes are shown in blue, while special vertex-centered function are highlighted in red.

graph <- gets.graph.tadpole(h = 1/4)
graph_to_get_loc <- gets.graph.tadpole(h = 1/40)
loc <- graph_to_get_loc$get_mesh_locations()


A <- graph$fem_basis(loc)
A <- cbind(A, A[, 1]*0)  # this plots the graph

x <- graph_to_get_loc$mesh$V[, 1]
y <- graph_to_get_loc$mesh$V[, 2]
x <- plotting.order(x, graph_to_get_loc)
y <- plotting.order(y, graph_to_get_loc)

x_range <- range(x)
y_range <- range(y)
z_range <- c(0,1)

# Start plot
p <- plot_ly()

# Add function lines
for (i in 1:ncol(A)) {
  z <- plotting.order(A[, i], graph_to_get_loc)
  mycolor <- if (i == ncol(A)) "black" else "blue"
  if (i %in% c(1,2)) mycolor <- "red"
  # Add basis function trace
  p <- add_trace(p, x = x, y = y, z = z, type = "scatter3d", mode = "lines",
                 line = list(color = mycolor, width = 3), showlegend = FALSE)
  
  # Add vertical segments from (x, y, 0) to (x, y, z)
  for (j in 1:length(x)) {
    p <- add_trace(p,
      x = rep(x[j], 2),
      y = rep(y[j], 2),
      z = c(0, z[j]),
      type = "scatter3d",
      mode = "lines",
      line = list(color = "gray", width = 1),
      showlegend = FALSE
    )
  }
}

z <- plotting.order(A[, 1], graph_to_get_loc)
z_new <- rep(NA, length(z))
z_new[1:11] <- 0
p <- add_trace(p, x = x, y = y, z = z_new, type = "scatter3d", mode = "lines",
                 line = list(color = "green", width = 3), showlegend = FALSE)

z <- plotting.order(A[, 2], graph_to_get_loc)
z_new <- rep(NA, length(z))
z_new[c(31:51, 111:121)] <- 0
p <- add_trace(p, x = x, y = y, z = z_new, type = "scatter3d", mode = "lines",
                 line = list(color = "green", width = 3), showlegend = FALSE)
# Display plot
p %>% layout(
    scene = global.scene.setter(x_range, y_range, z_range, z_aspectratio = 1))

Figure 1: Illustration of the system of basis functions \(\{\varphi_j^e, \phi_v\}\) on the tadpole graph. Notice that the sets \(\mathcal{N}_{v}\) are depicted in green and their corresponding basis functions are shown in red.

Having introduced the system of basis functions \(\{\varphi_j^e, \phi_v\}\), which will be referred to as \(\{\psi_j\}_{j=1}^{N_h}\), we can now define the finite element space \(V_h\subset H^1(\Gamma)\) as \(V_h = \left(\bigoplus_{e\in\mathcal{E}} V_{h_e}\right)\bigoplus V_v\), where \(V_v = \text{span}\left(\{\phi_v:v\in\mathcal{V}\}\right)\) and \(\dim\left(V_h\right)\) is given by \(N_h = |\mathcal{V}| + \sum_{e\in\mathcal{E}}n_e\).

2 Eigenfunctions and eigenvalues on the tadpole graph

Let \(\Gamma_T = (\mathcal{V},\mathcal{E})\) characterize the tadpole graph with \(\mathcal{V}= \{v_1,v_2\}\) and \(\mathcal{E}= \{e_1,e_2\}\). The left edge \(e_1\) has length 1 and the circular edge \(e_2\) has length 2. As discussed before, a point on \(e_1\) is parameterized via \(s=\left(e_1, t\right)\) for \(t \in[0,1]\) and a point on \(e_2\) via \(s=\left(e_2, t\right)\) for \(t\in[0,2]\). One can verify that \(-\Delta_\Gamma\) has eigenvalues \(0,\left\{(i \pi / 2)^2\right\}_{i \in \mathbb{N}}\) and \(\left\{(i \pi / 2)^2\right\}_{2 i \in \mathbb{N}}\) with corresponding eigenfunctions \(\phi_0\), \(\left\{\phi_i\right\}_{i \in \mathbb{N}}\), and \(\left\{\psi_i\right\}_{2 i \in \mathbb{N}}\) given by \(\phi_0(s)=1 / \sqrt{3}\) and \[\begin{equation*} \phi_i(s)=C_{\phi, i}\begin{cases} -2 \sin (\frac{i\pi}{2}) \cos (\frac{i \pi t}{2}), & s \in e_1, \\ \sin (i \pi t / 2), & s \in e_2, \end{cases}, \quad \psi_i(s)=\frac{\sqrt{3}}{\sqrt{2}} \begin{cases} (-1)^{i / 2} \cos (\frac{i \pi t}{2}), & s \in e_1, \\ \cos (\frac{i \pi t}{2}), & s \in e_2, \end{cases}, \end{equation*}\] where \(C_{\phi, i}=1\) if \(i\) is even and \(C_{\phi, i}=1 / \sqrt{3}\) otherwise. Moreover, these functions form an orthonormal basis for \(L_2(\Gamma_T)\).

graph <- gets.graph.tadpole(h = 0.01)
eigenpairs <- gets.eigen.params(N_finite = 25, kappa = 1, alpha = 0.5, graph)
EIGENFUN <- eigenpairs$EIGENFUN
EIGENVAL <- eigenpairs$EIGENVAL
INDEX <- eigenpairs$INDEX

graph.plotter.3d(graph, INDEX, EIGENVAL, EIGENFUN)

Figure 2: Illustration of the eigenfunctions on the tadpole graph.

3 Projection onto a fine space-time mesh

3.1 Temporal piecewise projection

The piecewise constant projection \(U_{\text{piecewise}}(t^*_\ell)\) of the approximated values \(U_{\text{approx}}(t_k)\), defined on a coarse time grid \(\{t_k\}_{k=0}^{N}\), onto a finer grid \(\{t^*_\ell\}_{\ell=0}^{M}\), is given by

\[\begin{equation} U_{\text{piecewise}}(t^*_\ell) = \begin{cases} U_{\text{approx}}(t_0), & \text{if } t^*_\ell = 0 \\\\ U_{\text{approx}}(t_k), & \text{if } t^*_\ell \in (t_{k-1}, t_k], \quad \text{for } k = 1, \dots, N. \end{cases} \end{equation}\]

This defines a function that is constant on each interval \((t_{k-1}, t_k]\), and takes the value of the approximation at the right endpoint \(t_k\) of the interval.

See the Fuctionality page for the implementation of the function construct_piecewise_projection() that performs this operation.

3.2 Spatial projection

A function \(U_{\text{coarse}}(s)\) defined on a coarse mesh with \(N_{h}\) nodes can be projected onto a fine mesh with \(N_{h_{\text{ok}}}\) nodes by doing \(U_{\text{fine}}(s) = \boldsymbol{\Psi} U_{\text{coarse}}(s)\), where \(\boldsymbol{\Psi}\) is a matrix with entries \(\boldsymbol{\Psi}_{ij}=\psi^j_h(s_i)\) for \(j=1\dots, N_h\) and \(i = 1,\dots N_{h_{\text{ok}}}\).

# Parameters
T_final <- 2
kappa <- 15
N_finite = 4 # choose even
adjusted_N_finite <- N_finite + N_finite/2 + 1
# Coefficients for u_0 and f
coeff_U_0 <- 50*(1:adjusted_N_finite)^-1
coeff_U_0[-5] <- 0
coeff_FF <- rep(0, adjusted_N_finite)
coeff_FF[7] <- 10

AAA = 1
OMEGA = pi

# Overkill parameters
overkill_time_step <- 0.01
overkill_h <- 0.01

# Finest time and space mesh
overkill_time_seq <- seq(0, T_final, length.out = ((T_final - 0) / overkill_time_step + 1))
overkill_graph <- gets.graph.tadpole(h = overkill_h)

# Compute the weights on the finest mesh
overkill_graph$compute_fem() # This is needed to compute the weights
overkill_weights <- overkill_graph$mesh$weights


alpha <- 0.5 # from 0.5 to 2
beta <- alpha / 2

# Compute the eigenvalues and eigenfunctions on the finest mesh
overkill_eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = overkill_graph)
EIGENVAL_ALPHA <- overkill_eigen_params$EIGENVAL_ALPHA # Eigenvalues (they are independent of the meshes)
overkill_EIGENFUN <- overkill_eigen_params$EIGENFUN # Eigenfunctions on the finest mesh

# Compute the true solution on the finest mesh
overkill_U_true <- overkill_EIGENFUN %*% 
  outer(1:length(coeff_U_0), 
        1:length(overkill_time_seq), 
        function(i, j) (coeff_U_0[i] + coeff_FF[i] * G_sin(t = overkill_time_seq[j], A = AAA, lambda_j_alpha_half = EIGENVAL_ALPHA[i], omega = OMEGA)) * exp(-EIGENVAL_ALPHA[i] * overkill_time_seq[j]))


h <- 0.2
time_step <- 0.2
m <- 1
graph <- gets.graph.tadpole(h = h)
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
L <- kappa^2*C + G
eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = graph)
EIGENFUN <- eigen_params$EIGENFUN
U_0 <- EIGENFUN %*% coeff_U_0 # Compute U_0 on the current mesh
A <- graph$fem_basis(overkill_graph$get_mesh_locations())

time_seq <- seq(0, T_final, length.out = ((T_final - 0) / time_step + 1))
my_op_frac <- my.fractional.operators.frac(L, beta, C, scale.factor = kappa^2, m = m, time_step)
INT_BASIS_EIGEN <- t(overkill_EIGENFUN) %*% overkill_graph$mesh$C %*% A
# Compute matrix F with columns F^k
FF_approx <- t(INT_BASIS_EIGEN) %*% 
  outer(1:length(coeff_FF), 
        1:length(time_seq), 
    function(i, j) coeff_FF[i] * g_sin(r = time_seq[j], A = AAA, omega = OMEGA))

U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx[, 1] <- U_0
for (k in 1:(length(time_seq) - 1)) {
  U_approx[, k + 1] <- as.matrix(my.solver.frac(my_op_frac, my_op_frac$C %*% U_approx[, k] + time_step * FF_approx[, k + 1]))
}

projected_U_approx <- A %*% U_approx
projected_U_piecewise <- construct_piecewise_projection(projected_U_approx, time_seq, overkill_time_seq)

graph.plotter.3d(overkill_graph, overkill_time_seq, overkill_time_seq, overkill_U_true, projected_U_piecewise)

Figure 3: Illustration of the projection of the approximated solution \(U_{\text{approx}}\) (called projected_U_piecewise) onto a fine space-time mesh. The true solution \(U_{\text{true}}\) (called overkill_U_true) is shown in red, while the projection is shown in blue.

4 References

grateful::cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.5.0 (R Core Team 2025) and the following R packages: gsignal v. 0.3.7 (Van Boxtel, G.J.M., et al. 2021), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), knitr v. 1.50 (Xie 2014, 2015, 2025), Matrix v. 1.7.3 (Bates, Maechler, and Jagan 2025), MetricGraph v. 1.5.0.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), patchwork v. 1.3.1 (Pedersen 2025), plotly v. 4.10.4 (Sievert 2020), RColorBrewer v. 1.1.3 (Neuwirth 2022), renv v. 1.0.7 (Ushey and Wickham 2024), reshape2 v. 1.4.4 (Wickham 2007), rmarkdown v. 2.29 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2024), rSPDE v. 2.5.1.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.4.0 (Wickham, Pedersen, and Seidel 2025), tidyverse v. 2.0.0 (Wickham et al. 2019), viridisLite v. 0.4.2 (Garnier et al. 2023), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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———. 2023b. “Statistical Inference for Gaussian Whittle-Matérn Fields on Metric Graphs.” arXiv Preprint arXiv:2304.10372. https://doi.org/10.48550/arXiv.2304.10372.

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